OP-01005 Open

A mixing action whose crossed product has more than one Cartan subalgebra up to unitary conjugacy

Problem. Is there a free mixing probability measure preserving action $\Gamma \curvearrowright (X, \mu)$ of a nonamenable group $\Gamma$ such that $L^\infty(X) \rtimes \Gamma$ does not have a unique Cartan subalgebra up to unitary conjugacy?

Source: S. Popa, Some open problems in W*-rigidity (2013); listed as C.2 in J. Peterson, Open problems in operator algebras (version of 15 May 2026).

Imported from math.uwaterloo.ca/~j37peter/problems.html.

Classification

Primary: OA — Operator Algebras
Secondary: DS, GR
Keywords: Cartan subalgebras, group measure space construction, mixing actions, Peterson's operator algebra problems
Literature checked: Oct. 26, 2026 by clemens-admin

References

@misc{popa2013problems-c2,
  author       = {Popa, Sorin},
  title        = {Some open problems in {W}$^*$-rigidity},
  howpublished = {Slides, Paris, June 2013},
  year         = {2013},
  url          = {https://www.math.ucla.edu/~popa/ProblemsJune2013.pdf},
}
@misc{peterson-c2,
  author       = {Peterson, Jesse},
  title        = {Open problems in operator algebras},
  howpublished = {Web page},
  year         = {2026},
  url          = {https://www.math.uwaterloo.ca/~j37peter/problems.html},
  note         = {Problem C.2; version of 15 May 2026},
}
@article{connesjones1982,
  author  = {Connes, Alain and Jones, Vaughan},
  title   = {A {II}$_1$ factor with two nonconjugate {C}artan subalgebras},
  journal = {Bull. Amer. Math. Soc.},
  volume  = {6},
  number  = {2},
  pages   = {211--212},
  year    = {1982},
  doi     = {10.1090/S0273-0979-1982-14981-3},
}
@article{ozawapopa2010ii,
  author  = {Ozawa, Narutaka and Popa, Sorin},
  title   = {On a class of {II}$_1$ factors with at most one {C}artan subalgebra, {II}},
  journal = {Amer. J. Math.},
  volume  = {132},
  number  = {3},
  pages   = {841--866},
  year    = {2010},
  doi     = {10.1353/ajm.0.0121},
}
@article{popavaes2010,
  author  = {Popa, Sorin and Vaes, Stefaan},
  title   = {Group measure space decomposition of {II}$_1$ factors and {W}$^*$-superrigidity},
  journal = {Invent. Math.},
  volume  = {182},
  number  = {2},
  pages   = {371--417},
  year    = {2010},
  doi     = {10.1007/s00222-010-0268-5},
}
@article{speelman2012,
  author  = {Speelman, An and Vaes, Stefaan},
  title   = {A class of {II}$_1$ factors with many non conjugate {C}artan subalgebras},
  journal = {Adv. Math.},
  volume  = {231},
  number  = {3-4},
  pages   = {2224--2251},
  year    = {2012},
  doi     = {10.1016/j.aim.2012.07.004},
}

Citing this problem

This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.

OP-01005, "A mixing action whose crossed product has more than one Cartan subalgebra up to unitary conjugacy", clopen.solutions.